|Over the past decade, much
debate has arisen between mathematicians and
mathematics educators. These debates have significantly distracted the
attention of key players at all levels, and have impeded efforts to
improve mathematics learning in this country. This document represents
an attempt to identify a preliminary list of positions on which many
may be able to agree.
Our effort arose out of discussions between Richard Schaar and major
players in both communities. He suspected that some of these
disagreements might be more matters of language and lack of
communication than representative of fundamental differences of view.
To test this idea, he convened a small group of mathematicians and
We tried to bring clarity to key perspectives on K-12 mathematics
education. We began by exploring typical "flashpoint" topics and probed
our own positions on each of these to determine whether and where we
agreed or disagreed. For the first meeting, held in December 2004, we
began with summary statements drawn from prior exchanges among the
members of our group. We affirmed some agreements in this meeting, and
"discovered" others. We listened closely to one another, frequently
asking for clarification, or for examples. We tested our understanding
of others' points of view by proposing statements that we then examined
collectively. We drafted this document as a group, composing actual
text as we worked. One of us typed, and our emerging draft was
projected onto a screen in the meeting room. The process enabled us to
take issue with particular words and terms, and then reshape them until
all of us were satisfied. We were forced to look closely at our own
language and to seek common ground, not only in the terms we used, but
even in their nuanced meaning.
This document was completed at our second meeting, in June 2005. All of
us are encouraged by the extent of our agreements. The document treats
only a subset of the controversial issues, many of which arise in K-8
mathematics. We expect to continue the process by examining a wider
range of major issues in mathematics education. We have necessarily
limited ourselves to questions depending primarily on disciplinary
judgment, as opposed to those requiring empirical evidence.
We begin with three fundamental assertions and continue with a list of
areas in which we found common ground. For each, we have written a
short paragraph that captures the fundamental points of our agreement.
Our next step is to explore how others respond to the document, and to
use their responses to decide how best to make progress on the aims of
this project. Our goal is to forge new alliances, across communities,
necessary to develop effective solutions to the serious problems that
plague mathematics education in this country.
All students must have a solid grounding in mathematics to function
effectively in today's world. The need to improve the learning of
traditionally underserved groups of students is widely recognized;
efforts to do so must continue. Students in the top quartile are
underserved in different ways; attention to improving the quality of
their learning opportunities is equally important. Expectations for all
groups of students must be raised. By the time they leave high school,
a majority of students should have studied calculus.
- Basic skills with numbers continue to be vitally important
variety of everyday uses. They also provide crucial foundation for the
higher-level mathematics essential for success in the workplace which
must now also be part of a basic education. Although there may have
been a time when being to able to perform extensive paper-and-pencil
computations mechanically was sufficient to function in the workplace,
this is no longer true. Consequently, today's students need proficiency
with computational procedures. Proficiency,
as we use the term, includes both computational fluency and
understanding of the underlying mathematical ideas and principles.2
- Mathematics requires careful reasoning about precisely
defined objects and concepts. Mathematics is communicated by
means of a powerful language whose vocabulary must be learned. The
to reason about and justify mathematical statements is fundamental,
as is the ability to use terms and notation with appropriate
degrees of precision. By precision,
we mean the use of terms and symbols, consistent with mathematical
definitions, in ways appropriate for students at particular grade
levels. We do not mean
formality for formality's sake.
- Students must be able to formulate and solve problems.
Mathematical problem solving includes being able to (a) develop a clear
understanding of the problem that is being posed; (b) translate the
problem from everyday language into a precise mathematical question;
(c) choose and use appropriate methods to answer the question; (d)
interpret and evaluate the solution in terms of the original problem,
and (e) understand that not all questions admit mathematical solutions
and recognize problems
that cannot be solved mathematically.
Discussions of the following items are often riddled with difficulties
in communication, making it sometimes confusing to determine whether
how much disagreement exists. Issues also arise from a confounding of a
mathematical idea with its implementation in the classroom. For
example, the fact that algorithms have often been taught badly does not
imply that algorithms themselves are bad. We worked to clarify issues
and terms and arrived at statements with which we agreed.
- Automatic recall
of basic facts: Certain
procedures and algorithms in mathematics are so basic and have such
wide application that they should be practiced to the point of
automaticity. Computational fluency in whole number arithmetic is
vital. Crucial ingredients of computational fluency are
efficiency and accuracy. Ultimately, fluency requires automatic recall
basic number facts: by basic number
facts, we mean addition
and multiplication combinations of integers 0 -- 10. This goal can be
accomplished using a variety of instructional methods.
Calculators can have a useful role even in the lower
grades, but they must be used carefully, so as
not to impede the acquisition of fluency with basic facts and
computational procedures. Inappropriate use of calculators may
also interfere with students' understanding of the meaning of
fractions and their ability to compute with fractions. Along the same
lines, graphing calculators can enhance students' understanding of
functions, but students must develop a sound idea of what graphs
are and how to use them independently of the use of a graphing
algorithms: Students should be able to
use the basic algorithms of whole number arithmetic fluently, and
they should understand how and why the algorithms work. Fluent use
and understanding ought to be developed concurrently. These basic
algorithms were a major intellectual accomplishment. Because they
embody the structure of the base-ten number system, studying them can
reinforce students' understanding of the place value system.
More generally, an algorithm
is a systematic procedure
involving mathematical operations that uses a finite number of steps to
produce a definite
answer. An algorithm can be implemented in different ways; different
recording methods for the same algorithm do not constitute different
The idea of an algorithm is fundamental in mathematics. Studying
algorithms beyond those of whole number arithmetic provides
students to appreciate the diversity and importance of algorithms.
include constructing the bisector of an angle; solving two linear
two unknowns; calculating the square root of a number by a succession
of dividing and averaging.
Understanding the number meaning of fractions is
critical. Ratios, proportions, and percentages
cannot be properly understood without fractions. The arithmetic of
fractions is important as a foundation for algebra.
mathematics in "real world" contexts:
It can be helpful to motivate and introduce mathematical ideas
through applied problems. However, this approach should not be elevated
to a general principle. If all school mathematics is taught using real
world problems, then some important topics may not receive
adequate attention. Teachers must choose contexts with care. They need
to manage the use of real-world problems or mathematical
applications in ways that focus students' attention on the mathematical
ideas that the problems are intended to develop.
methods: Some have suggested the exclusive use of
small groups or discovery learning at the
expense of direct instruction in teaching mathematics. Students can
learn effectively via a mixture of direct instruction, structured
investigation, and open exploration. Decisions about what is
better taught through direct instruction and what might be better
taught by structuring explorations for students should be made on the
basis of the particular mathematics, the goals for learning, and the
students' present skills and knowledge. For example, mathematical
conventions and definitions should not be taught by pure discovery.
Correct mathematical understanding and conclusions are the
responsibility of the teacher. Making good decisions about
the appropriate pedagogy to use depends on teachers having solid
knowledge of the subject.
- Teacher knowledge:
effectively depends on a solid understanding of the material. Teachers
be able to do the mathematics they are teaching, but that is not
sufficient knowledge for teaching. Effective teaching requires an
understanding of the underlying meaning and justifications for
the ideas and procedures to be taught, and the ability to make
connections among topics. Fluency, accuracy, and precision in the use
of mathematical terms and symbolic notation are also crucial. Teaching
demands knowing appropriate representations
for a particular mathematical idea, deploying these with precision, and
bridging between teachers' and students' understanding. It
requires judgment about how to reduce mathematical complexity and
manage precision in ways that make the mathematics accessible to
students while preserving its integrity.
Well-designed instructional materials, such as textbooks,
teachers' manuals and software, may provide significant mathematical
support, but cannot substitute for highly qualified, knowledgeable
teachers. Teachers' mathematical knowledge must be developed through
solid initial teacher preparation and ongoing, systematic professional
grateful to the National Science Foundation and Texas Instruments Inc.
for funding this portion of our work
2Kilpatrick, J., Swafford, J. and Findell, B. (Eds.). Adding It Up: Helping Children Learn
Mathematics, Washington, DC: National Academy Press, 2001.